Chapter 4
77
Test Form A
Name
__________________________________________
Date
Chapter 4
Class
__________________________________________
Section _______________________
1. Evaluate the integral:
12 t
3 43
t C
4
(b)
(d)
1
C
3t 23
(e) None of these
3
2
C
(d) 5sec3 x sec x tan2 x C
(e) None of these
(a) x3 3x C
(a) cos C
(d)
1 2
sin C
2
(c)
1
5
(c)
x3
xC
3
(c)
cos 3 3 cos2 2 sin2 1 cos2 sec3 x tan x C
x3 x
dx.
x
(b) 2x C
2x3 x 1
x2
4. Evaluate the integral:
3 23
t C
2
5 sec x tan x dx.
(b) 5 sec x C
(d)
(c)
(a) 5 sec3 x tan x C
3. Evaluate the integral:
____________________________
3 t dt.
(a)
2. Evaluate the integral:
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Test Bank
(e) None of these
sin3 d.
1 cos2 (b) cos C
(e) None of these
5. Find y f x if f x x2, f 0 7, and f 0 2.
1 4
12 x
(a) x2 9
(b)
7x 2
(d) x 4 84x 24
(e) None of these
(c) x2 7x 2
6. Use at 32 feet per second squared as the acceleration due to gravity. A ball is thrown vertically
upward from the ground with an initial velocity of 96 feet per second. How high will the ball go?
(a) 32 feet
(b) 64 feet
(d) 144 feet
(e) None of these
(c) 24 feet
78
Chapter 4
Integration
10
7. Use the properties of sigma notation and the summation formulas to evaluate the given sum:
i
i1
(a) 83
(b) 245
(d) 81
(e) None of these
8. Let s n 2
2i 3.
(c) 305
1 n
n
. Find the limit of sn as n → .
n
i
2
2
i1
(a)
17
12
(b)
(d)
20
3
(e) None of these
9. Let f x 10
3
x ≤ 3
. Use geometric formulas to find
x > 3
3,
6 x,
(a) 15
(b) 13
(d) 30
(e) None of these
(c)
14
3
5
f x dx.
0
(c) 11
4
10. Use the Fundamental Theorem of Calculus to evaluate the integral:
x dx.
1
14
(b) 3
(a) 1
(d)
14
3
(c) 7
(e) None of these
11. Find the average value of f x 2x2 3 on the interval 0, 2.
22
3
(b)
(d) 27
17
3
(c) 4
(e) None of these
12. Evaluate the integral:
(a)
1 3
x 57 C
21
(d)
x3 x 4
5x
3 4
6
x2x 3 56 dx.
(b)
C
(c)
x3x3 57
C
21
(c)
1
2
(e) None of these
2
13. Evaluate the integral:
1 3
x 57 C
7
x 1 dx.
0
(a) 0
(b) 1
(d) 2
(e) None of these
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(a)
Chapter 4
14. Evaluate the integral:
cos 3x dx.
(a) sin 3x C
(d)
1
3
(b) sin 3x C
sin 3x C
15. Evaluate the integral:
(e) None of these
x1 x dx.
2 3x
C
21 x
(a) x2
1 x32 C
3
(b)
(d) 2
2 3x1 x32 C
15
(e) None of these
3
16. Use Simpson’s Rule with n 4 to approximate
2
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3
(c) sin 2x2 C
(c)
x2
1 x32 C
3
1
dx.
x 12
(a) 0.5004
(b) 0.5090
(d) 1.7396
(e) None of these
(c) 2.5000
Test Bank
79
Chapter 4
Integration
Test Form B
Name
__________________________________________
Date
Chapter 4
Class
__________________________________________
Section _______________________
1. Evaluate the integral:
5 x2 dx.
x3 C
3
(a)
(d) (b)
1
C
4x8
(e) None of these
(a)
1
3
(d)
13
csc3
(b) 6 csc2 x cot x C
(c) 3 cot x C
(e) None of these
x4 x3
dx.
x2
(a) 2x3 3x2 C
(a)
1
C
9x9
3 csc2 x dx.
xC
(b)
3
4x2 5x C
20
4. Evaluate the integral:
(c) csc2 x C
3. Evaluate the integral:
(d)
5 75
x C
7
5
2. Evaluate the integral:
____________________________
x3 x2
C
3
2
(c) 2x 1 C
(e) None of these
sec3 tan d.
1 tan2 1
sec4 C
4
(d) sec C
(b)
1
sec2 C
2
(c)
1
sec2 tan2 C
4
(e) None of these
5. Find y f x if f x x 2, f 0 3, f 0 1.
x3
2x2 C
6
(a)
1 3
x x2 3x 1
6
(b)
(d)
1 3
21
61
x x2 x C
6
2
6
(e) None of these
(c) x3 6x2 18x 6
6. Use a t 32 feet per second squared as the acceleration due to gravity. A ball is thrown
vertically upward from the ground with an initial velocity of 56 feet per second. For how many
seconds will the ball be going upward?
(a) 134 sec
(b) 24 sec
(d) 1 sec
(e) None of these
(c)
7
8
sec
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80
Chapter 4
Test Bank
10
7. Use the properties of sigma notation and the summation formulas to evaluate the given sum:
i
2
81
3i 2.
i1
(a) 530
(b) 128
(d) 915
(e) None of these
8. Let s n n
1
i1
i
n
(c) 126
1n. Find the limit of sn as n → .
2
(a)
5
3
(b)
(d)
17
24
(e) None of these
9. Let f x (c)
x ≤ 3
. Use geometric formulas to find
x > 3
3,
6 x,
(a) 18
7
3
(b) 15
27
2
(c)
6
f x dx.
1
(d)
21
2
(e) None of these
2
10. Use the Fundamental Theorem of Calculus to evaluate the integral:
1
1
(a) C
x
(d) (b) 3
2
10
3
3
4
1
dx.
x2
(c)
1
2
(e) None of these
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11. Find the average value of f x 3x2 2 on the interval 0, 2.
(a) 1
(b) 5
(c) 6
(d) 2
(e) None of these
12. Evaluate the integral: x x2 14 dx.
(a)
(d)
1 2
2
5
10 x x 1
1 2
5
5 x 1 C
13. Evaluate the integral:
(a)
(d)
1
4
2
8 sin 3x cos 3x
1
4
12 sin 3x C
(b)
1 2
10 x
15 C
1
(c) 5x3 x5 C
(e) None of these
sin3 3x cos 3x dx.
C
(b)
1
4
sin4 3x C
(e) None of these
(c) 3 sin2 3x 3 cos2 3x sin2 3x C
82
Chapter 4
Integration
1
14. Evaluate the integral:
x dx.
1
(a) 1
(b) 0
(c) 2
(d) 1
(e) None of these
15. Evaluate the integral: xx 1 dx.
(a)
3x 2
C
2x 1
(b)
(d)
1 2
x x 132 C
3
(e) None of these
2
x 1323x 2 C
15
2
1
x 1x 1 C
4
1
dx.
x 12
(a) 0.5004
(b) 2.5000
(d) 1.7396
(e) None of these
(c) 0.5090
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3
16. Use Trapezoidal Rule with n 4 to approximate
(c)
Chapter 4
Test Bank
Test Form C
Name
__________________________________________
Date
Chapter 4
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator is needed for some problems.
1. Evaluate the integral:
ax b dx.
(a)
ab 2
x C
2
(b) a C
(d)
a 2
x bx
2
(e) None of these
2. Evaluate the integral:
(a)
(c)
a 2
x bx C
2
(c)
3 32
x
2x2 C
2
3 4x32
dx.
x
3
x 2x2 C
2
(d) 6x 2x2 C
3
(b) x32 4 C
2
(e) None of these
3. Find the particular solution of the equation fx 2x12 that satisfies the condition f 1 6.
(a) f x 4x 2
(b) f x 4x C
(d) f x 2x 4
(e) None of these
(c) f x 2x 6 2
2
4. The rate of growth of a particular population is given by
dP
50t 2 100t 32,
dt
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83
where P is the population size and t is the time in years. The initial population is 25,000. Use a graphing
calculator to graph the population function. Then use the graph to estimate how many years it will take
for the population to reach 50,000.
(a) 15.7
(b) 14.5
(d) 38.4
(e) None of these
(c) 2
5. An object has a constant acceleration of 42 feet per second squared, an initial velocity of 18 feet
per second, and an initial position of 3 feet. Find the position function describing the motion of
this object.
(a) s 42t 2 3
(b) s 42t 2 18t 3
(d) s 21t 2 18t 3
(e) None of these
(c) s 21t 2 5
84
Chapter 4
Integration
6. Identify the sum that does not equal the others.
5
(a)
6
3n 1
(b)
n0
3k 2
8
(c)
k1
3j 8
j3
6
(d)
i 2
(e) None of these
i1
7. Use a graphing calculator to graph f x 2x3 3x2 5. Then use the upper sums to approximate
the area of the region between the x-axis and f on the interval 0, 1 using 4 subintervals.
(a) 2.81
(b) 4.06
(c) 3.5
(d) 3
(e) None of these
8. Determine the interval(s) on which f x x 2 is integrable.
(a) 0, 5
(b) 3, 0
(d) a, b, and c
(e) a and c
(c) 1, 4
9. Use a graphing calculator as an aid in finding the area of the region above the x-axis bounded by
f x 5x3 35x 30.
(c) 3.75
(a) 10
(b) 160
(d) 55
(e) None of these
1
10. Consider F x 1 t 2 dt. Find Fx.
x
(d) 1 x2
(e) None of these
2
x
(b)
11. Choose the correct quantity to fill in the blank:
1 x2
______________ dx ax2 a2 4 C.
(a) ax2 a23
(b) 4ax2 a23
(d) 4aax2 a23
(e) None of these
12. Evaluate the integral:
(a)
1 3
6x
sec3 x2 C
(d) tan x3 C
(c) 1
(c) 15ax2 a25
x sec2x2 dx.
(b)
1
2
tan x2 C
(e) None of these
(c)
1 2
2x
tan x2 C
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1
(a) 1 x2
Chapter 4
Test Bank
13. Use a graphing calculator as an aid in finding the area in the second quadrant bounded by the x-axis
x8 1.
and f x x2
3
(a)
1
4
(b)
8
2.667
3
(d)
16
1 232
9
(e) None of these
14. Use the general power rule to evaluate the integral:
(b) (d) 4
9 5x232 C
15
(e) None of these
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5x
x 2
(c)
2
9 5x232 C
3
x9 5x2dx.
1
9 5x232 C
10
16
1.778
9
(a) 15. Evaluate the integral:
(c)
1
9 5x232 C
15
dx.
(a)
2
x 2 x 4 C
3
(b)
5
x 4 lnx 2 C
2
(d)
10
x 2 x 4 C
3
(e) None of these
(c) 5x 4 lnx 2 C
85
86
Chapter 4
Integration
Test Form D
Name
__________________________________________
Date
Chapter 4
Class
__________________________________________
Section _______________________
1. Evaluate the integral:
2. Evaluate the integral:
3. Evaluate the integral:
4. Evaluate the integral:
x 3 dx.
x3 x2
dx.
x2
____________________________
3 csc x cot x dx.
cos3 d.
2 2 sin2 5. Find the function, y f x, if f x 2x 1 and f 1 3.
6. Use a t 32 feet per second squared as the acceleration due to gravity. An object is thrown
vertically downward from the top of a 480-foot building with an initial velocity of 64 feet per second.
With what velocity does the object hit the ground?
7. Let s n 1 n n
. Find the limit of sn as n → .
n
2i
2
i1
8. Write the definite integral that represents the area of the region enclosed by y 4x x2 and the x-axis.
d
dx
x
2t2 52 dt.
2
1
10. Use the Fundamental Theorem of Calculus to evaluate
1
11. Find the average value of f x sin x on the interval
1
12. Evaluate the integral:
x1 x2 dx.
0
13. Evaluate the integral:
sec2 x
dx.
tan x
3
14. Evaluate the integral:
x 2 dx.
0
3 t 2 dt.
4 , 2 .
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9. Evaluate:
Chapter 4
15. Evaluate the indefinite integral:
x
x 1
dx.
2
16. Use Simpson’s Rule with n 4 to approximate
© Houghton Mifflin Company. All rights reserved.
1
1
dx.
x 12
Test Bank
87
88
Chapter 4
Integration
Test Form E
Name
__________________________________________
Date
Chapter 4
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator is needed for some problems.
1. Evaluate the integral:
2. Evaluate the integral:
ax2 bx3
dx.
x
x
dx.
x 13
3. Find the function, y f x, if f x 2x 1 and f 1 3.
4. The rate of growth of a particular population is given by
dP
40t3 70t 43,
dt
where P is the population size and t is the time in years. The initial population is 16,000.
a. Determine the population function.
b. Use a graphing calculator to graph the population function. Then use the graph to estimate the number
of years until the population reaches 20,000. (Round your answer to one decimal place.)
c. Use the population function to calculate the population after 8 years.
n
5. Find the limit: lim
i1
6. Evaluate the integral:
3n.
2
x 3 dx.
7. Use a graphing calculator to graph f x x 4 6x3 11x2 6x. Then use the upper sums to
approximate the area of the region in the first quadrant bounded by f and the x-axis using
4 subintervals. (Round your answer to three decimals places.)
8. Use a graphing calculator as an aid to sketch the region whose area is indicated by the integral:
2
x 3 4x 6 dx.
0
9. Determine if f x 5
is integrable on the interval 0, 2. Give a reason for your answer.
2x 3
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n→
3i 1
n
Chapter 4
10. Use a graphing calculator to graph f x cos x sin x. Calculate the area in the first
quadrant bounded by the x-axis, the y-axis, and f.
x
11. Consider F x 1
dt. Find Fx and F 2.
1
t4
2
12. Evaluate the integral:
cos x
dx.
x
3
13. Evaluate the integral:
sec2 x dx.
4
1
14. Evaluate the integral:
xx2 1 dx.
0
15. Consider the region bounded by the x-axis, the function f x 1 x3, x 1, and x 1.
a. Use Trapezoidal Rule, with n 4, to approximate the area of the region. (Round the answer
to three decimal places.)
© Houghton Mifflin Company. All rights reserved.
b. Use Simpson’s Rule, with n 4, to approximate the area of the region. (Round the answer
to three decimal places.)
Test Bank
89